Combinatorial state sum invariant from categories

Combinatorial state sum invariant from categories
Aristide Baratin
LPT Orsay - CPHT Ecole Polytechnique - IPhT Saclay
Paris-Nord, April 2011
0910.1542[hep-th]
0812.4969[math.QA]
hep-th/0611042I 3D: Ponzano-Regge Rep(SU(2)) , Turaev-Viro Rep(U (su(2))q
I 4D: Dijkgraaf-Witten ( nite groups), Ooguri Rep(SU(2)), Crane-Yetter Rep( U (su(2))q
I Models of 4d quantum gravity: Barrett-Crane, EPRL-FK
Simplicial set S of algebraic data, or states labeling each simplex.
@ : S( )!S( )i n n 1
Weights w: S( )!C give an amplitude to a staten
P Q
Partition function: Z = w(s())s2S
Introduction
State sum models
State sum model is a discrete functional integral on a triangulated manifold:
A. Baratin | State sum invariant from a 2-category Introduction 2/37I 3D: Ponzano-Regge Rep(SU(2)) , Turaev-Viro Rep(U (su(2))q
I 4D: Dijkgraaf-Witten ( nite groups), Ooguri Rep(SU(2)), Crane-Yetter Rep( U (su(2))q
I Models of 4d quantum gravity: Barrett-Crane, EPRL-FK
Weights w: S( )!C give an amplitude to a staten
P Q
Partition function: Z = w(s())s2S
Introduction
State sum models
State sum model is a discrete functional integral on a triangulated manifold:
Simplicial set S of algebraic data, or states labeling each simplex.
@ : S( )!S( )i n n 1
A. Baratin | State sum invariant from a 2-category Introduction 2/37I 3D: Ponzano-Regge Rep(SU(2)) , Turaev-Viro Rep(U (su(2))q
I 4D: Dijkgraaf-Witten ( nite groups), Ooguri Rep(SU(2)), Crane-Yetter Rep( U (su(2))q
I Models of 4d quantum gravity: Barrett-Crane, EPRL-FK
P Q
Partition function: Z = w(s())s2S
Introduction
State sum models
State sum model is a discrete functional integral on a triangulated manifold:
Simplicial set S of algebraic data, or states labeling each simplex.
@ : S( )!S( )i n n 1
Weights w: S( )!C give an amplitude to a staten
A. Baratin | State sum invariant from a 2-category Introduction 2/37I 3D: Ponzano-Regge Rep(SU(2)) , Turaev-Viro Rep(U (su(2))q
I 4D: Dijkgraaf-Witten ( nite groups), Ooguri Rep(SU(2)), Crane-Yetter Rep( U (su(2))q
I Models of 4d quantum gravity: Barrett-Crane, EPRL-FK
Introduction
State sum models
State sum model is a discrete functional integral on a triangulated manifold:
Simplicial set S of algebraic data, or states labeling each simplex.
@ : S( )!S( )i n n 1
Weights w: S( )!C give an amplitude to a staten
P Q
Partition function: Z = w(s())s2S
A. Baratin | State sum invariant from a 2-category Introduction 2/37Introduction
State sum models
State sum model is a discrete functional integral on a triangulated manifold:
Simplicial set S of algebraic data, or states labeling each simplex.
@ : S( )!S( )i n n 1
Weights w: S( )!C give an amplitude to a staten
P Q
Partition function: Z = w(s())s2S
I 3D: Ponzano-Regge Rep(SU(2)) , Turaev-Viro Rep(U (su(2))q
I 4D: Dijkgraaf-Witten ( nite groups), Ooguri Rep(SU(2)), Crane-Yetter Rep( U (su(2))q
I Models of 4d quantum gravity: Barrett-Crane, EPRL-FK
A. Baratin | State sum invariant from a 2-category Introduction 2/37I ‘Metric’ models: explicit data on the edges of the triangulation ?
Why state sum invariants?
Combinatorial construction of manifold invariants, TQFT’s
Models of quantum geometry:
I Triangulation independent models of quantum geometry ?
Issue tied to di eomorphism symmetry
Introduction
State sum invariants
Constructing manifold invariants:
Philosophy: use combinatorics of the local ’Pachner moves’ of the triangulation to
convert a topological problem into an algebraic one.
A. Baratin | State sum invariant from a 2-category Introduction 3/37 Combinatorial construction of manifold invariants, TQFT’s
Models of quantum geometry:
I Triangulation independent models of quantum geometry ?
Issue tied to di eomorphism symmetry
I ‘Metric’ models: explicit data on the edges of the triangulation ?
Introduction
State sum invariants
Constructing manifold invariants:
Philosophy: use combinatorics of the local ’Pachner moves’ of the triangulation to
convert a topological problem into an algebraic one.
Why state sum invariants?
A. Baratin | State sum invariant from a 2-category Introduction 3/37 Models of quantum geometry:
I Triangulation independent models of quantum geometry ?
Issue tied to di eomorphism symmetry
I ‘Metric’ models: explicit data on the edges of the triangulation ?
Introduction
State sum invariants
Constructing manifold invariants:
Philosophy: use combinatorics of the local ’Pachner moves’ of the triangulation to
convert a topological problem into an algebraic one.
Why state sum invariants?
Combinatorial construction of manifold invariants, TQFT’s
A. Baratin | State sum invariant from a 2-category Introduction 3/37Introduction
State sum invariants
Constructing manifold invariants:
Philosophy: use combinatorics of the local ’Pachner moves’ of the triangulation to
convert a topological problem into an algebraic one.
Why state sum invariants?
Combinatorial construction of manifold invariants, TQFT’s
Models of quantum geometry:
I Triangulation independent models of quantum geometry ?
Issue tied to di eomorphism symmetry
I ‘Metric’ models: explicit data on the edges of the triangulation ?
A. Baratin | State sum invariant from a 2-category Introduction 3/37